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J. R. Soc. Interface (2010) 7, 651–665
doi:10.1098/rsif.2009.0371
Published online 14 October 2009
The control systems structures
of energy metabolism
Mathieu Cloutier and Peter Wellstead*
Hamilton Institute, National University of Ireland, Maynooth, Ireland
The biochemical regulation of energy metabolism (EM) allows cells to modulate their energetic
output depending on available substrates and requirements. To this end, numerous biomolecular mechanisms exist that allow the sensing of the energetic state and corresponding
adjustment of enzymatic reaction rates. This regulation is known to induce dynamic systems
properties such as oscillations or perfect adaptation. Although the various mechanisms of
energy regulation have been studied in detail from many angles at the experimental and theoretical levels, no framework is available for the systematic analysis of EM from a control systems
perspective. In this study, we have used principles well known in control to clarify the basic
system features that govern EM. The major result is a subdivision of the biomolecular mechanisms of energy regulation in terms of widely used engineering control mechanisms:
proportional, integral, derivative control, and structures: feedback, cascade and feed-forward
control. Evidence for each mechanism and structure is demonstrated and the implications for
systems properties are shown through simulations. As the equivalence between biological systems and control components presented here is generic, it is also hypothesized that our work
could eventually have an applicability that is much wider than the focus of the current study.
Keywords: control theory; energy metabolism; metabolic regulation;
systems biology
1. INTRODUCTION
Energy metabolism (EM) in this study will refer to
the metabolic reactions by which cells use available
substrates to produce energy in the form of ATP. Interestingly, these metabolic pathways are the most
preserved in living organisms; a fact that shows the
critical importance of EM in maintaining proper cellular function. Improper energy production and
regulation is reported in many diseases such as neurodegeneration (Lotharius & Brundin 2002; Parihar &
Brewer 2007), cardiovascular insufficiency (Kensara
et al. 2006; Hardie 2008), immunological disorders
(Buttgereit et al. 2000) and cancer (Carew & Huang
2002; Pedersen 2007). Energy regulation is therefore
of great importance to both the healthy operation of cellular processes and disease conditions. A better
understanding of the systems aspects of energy regulation could thus have significant implications in the
analysis of various diseases as well as in the design of
therapies. For example, Seyfried & Mukherjee (2005)
report that a correct understanding of energy regulation
in cancer cells allowed the design of a nutritional regime
to arrest the growth of an inoperable brain tumour.
However, a precise regime, both in terms of quantity
and timing, was necessary for the approach to be efficient (Seyfried & Mukherjee 2005). This in turn
implies a need for an understanding of dynamic control
mechanisms involved in cancer cells. Unfortunately,
this nutritional regime was not further validated in
large-scale clinical studies. But its success, even
though limited in scale and possible validity, is
nevertheless insightful. Such findings illustrate the
1.1. General considerations about energy
in cellular processes
Energy is involved in most, if not all, cellular processes.
Cellular growth requires free energy for the synthesis
and organization of cellular components and the cell
wall. Maintenance processes such as protein disposal
and maintaining membrane integrity and ion concentrations also require energy. All signalling pathways
will include protein phosphorylation cascades, in
which phosphate is transferred from energy-rich
molecules. The most fundamental function of a living
organism—the transcription of its genome and further
protein synthesis—can account for a significant portion
(up to 40%) of the energy budget (Buttgereit et al.
2000).
In this regard, energy obtained from the breakdown
of substrates such as glucose (GLC) has to be transferred to endergonic reactions. Adenosine triphosphate
(ATP) is the main ‘energy currency’ in living organisms
and must be continuously available in order to maintain
the cell’s capacity to fulfil its function—e.g. processing
signals and producing adequate responses. Energy is
thus a central player in the complex system that is the
living cell.
*Author for correspondence ( [email protected]).
Electronic supplementary material is available at http://dx.doi.org/
10.1098/rsif.2009.0371 or via http://rsif.royalsocietypublishing.org.
Received 21 August 2009
Accepted 14 September 2009
651
This journal is q 2009 The Royal Society
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Control systems structures of EM
M. Cloutier and P. Wellstead
importance of considering the dynamics of a system,
and thus a general control framework, such as the one
presented here, might be relevant for future studies on
the implications of EM in disease.
1.2. Mechanisms for energy production
and regulation
Glycolysis is found in virtually all cell species. This
pathway can convert the most readily available sugars
(especially GLC) into pyruvate (PYR) while regenerating two molecules of ATP. PYR can in turn be
processed through oxidative phosphorylation (OP) to
further produce energy with a yield of around 30 ATP
per molecule of GLC. In eukaryotes and higher organisms, OP takes place in mitochondria, hence their
common label as the ‘energy generator’ of the cell. In
prokaryotes, OP is much less organized, with oxidative
reactions occurring sparsely in the cytosolic fluid and at
the cell membrane. These two pathways (glycolysis and
OP) and their regulation are the focus of this study.
Although glycolysis and OP are the main pathways
for energy generation, cells and tissues have other mechanisms for energy production and storage. For example,
phosphagen molecules, such as phosphocreatine (PCr),
can act as a temporary buffer during high-demand
periods. This mechanism was shown to be important
in muscle tissue under various conditions of energy
demand and metabolic regimes (Hochachka &
McClelland 1997). Although other functions have
been unveiled for PCr, such as spatial buffering
(Bessman & Geiger 1981), we will limit our analysis
here to its potential function as a temporal buffer.
Simulation results will, however, show that PCr is present in sufficient quantities to perform both functions
(spatial and temporal buffering).
Finally, GLC is not the only energy substrate
involved in glycolysis. Some cells can store it under
various forms, such as glycogen (in mammalian tissues)
or starch (in plants). These local reserves can then be
drawn upon during high-demand periods or during
periods of nutrient limitation. In this study, we will
not analyse these phenomena, rather we will simply
consider that nutrients (GLC and O2) are present in
sufficient concentrations. However, other substrates
can enter the cells at some branch points in EM. In
this context, lactate (LAC) has been usually considered
as an undesirable by-product of anaerobic glycolysis,
but is now believed to be an important player in EM
in the cerebral tissue (Pellerin & Magistretti 1994;
Schurr 2006). Specifically, LAC is easily converted to
PYR through the reverse action of lactate dehydrogenase (LDH) and thus can rapidly feed OP. In neurons,
this reaction can even occur inside the mitochondria
(Schurr 2006), which suggests that LAC, and not
PYR, might be the substrate for OP. As a comparison,
GLC has to go through the sequence of nine reactions of
glycolysis in order to produce PYR. As for the implication of LAC in EM regulation, it has been shown in
cultivated neurons that increasing extracellular LAC
concentration leads to higher ATP levels in cells
(Ainscow et al. 2002). In the work reported here, we
will consider a reaction scheme where LAC can be
J. R. Soc. Interface (2010)
employed as an energy substrate in a way that
complements GLC usage.
1.3. Systems properties and control in biology
Systems properties have been reported for EM. In
particular, glycolytic oscillations are observed in yeast
because of specific interactions between glycolytic components (Chance et al. 1964; Termonia & Ross 1981).
Coupling between energy production and utilization
(Heinrich & Schuster 1996; Fell 1997) is also a
common observation. Finally, energy homeostasis is
reported in many tissues, even in the presence of large
changes in energy demand (Hochachka & McClelland
1997). It is known that these properties are observed
because of the possible interactions between the components of EM, and each study offers specific reasons
for the observation of such properties. However,
although the biomolecular mechanisms behind EM
(in addition to its properties as a regulated system)
are very well known, no general framework is available
with which to analyse the functional links between the
mechanisms and the systems properties of EM.
It is known that biophysical processes can exhibit
properties similar to those of control systems components. For example, the regulation of reaction rates
by the state of a system has already been compared
with proportional action (Yi et al. 2000; Csete &
Doyle 2002; El-Samad et al. 2002). Integral feedback
by the accumulation of molecules is reported to be
involved in homeostasis (Yi et al. 2000; El-Samad
et al. 2002), and feed-forward control is part of one of
the most recurrent patterns in gene regulation networks
(Mangan & Alon 2003), as well as being involved in the
regulation of glycolytic intermediates (Bali & Thomas
2001). More recently, a study by Chandra et al.
(2009) showed that a simplified model of glycolysis
could be studied within a control theory framework.
These observations show that control properties are
inherent to many biological regulation mechanisms
and suggest that it would be desirable and instructive
to construct a general control framework within which
to analyse such systems.
1.4. Application to energy metabolism
In this study, a control framework will be developed for
the regulation of EM. Our aim is to reveal the essential
control features, and thus we use the language of a first
course in classical control systems and focus upon
revealing the most basic control mechanisms and structures; questions of nonlinearity and details of stability
are left to subsequent specialist studies. For consistency,
we will use one control text (Seborg et al. 1989) as our
main reference for the engineering equivalents of the
EM regulation features that we discuss.
As a focus for the discussion, we use a generic
mathematical model of glycolysis and OP and their regulation by the energetic state. This generic model will then
be used to delineate the contribution of each component in
terms of its control action. (In control terms, we consider
the manipulated variables to be the energy-producing
reactions and the controlled variable to be the ATP
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M. Cloutier and P. Wellstead
653
astrocyte–neuron
LAC shuttle
vLAC
feed-forward
energy supply
energy
demand
GLC
LAC
vHK
O2
vLDH
ADP
glycolysis
+
vPK
PYR
oxidative
phosphorylation
vPFK
ATPases
vOP
ATP
vPFK2
energetic state
‘sensing’
AMP
F26P
integrator of
energetic state
flux regulation proportional
to energetic state
transient energy
supply
intracellular volume
extracellular volume
vCK
PCr
ADP
Cr
ATP
Figure 1. A generic model for energy metabolism.
concentration.) Using different hypotheses for EM
regulation, we will consider the dynamic response of the
generic model to changes in energy demand and associate
these responses with the various control actions used in
EM. This will allow the specific forms of control to be
identified and analogies drawn with classical control.
The generic mathematical model is presented in the
following section.
2. MODELLING ENERGY METABOLISM
2.1. Model presentation
Figure 1 illustrates the structure of a generic mathematical
model for EM. A complete description of the model,
with mass balances, kinetic equations and hypotheses
is given in the electronic supplementary material. The
biochemical regulation of EM pathways was mainly
elucidated by the analysis of enzyme kinetics, and as
a result detailed mathematical models are available for
EM in various cell species and tissues (Korzeniewski &
Zoladz 2001; Chassagnole et al. 2002; Lambeth &
Kushmerick 2002; Holzhütter 2004; Aubert & Costalat
2005; Cloutier et al. in press, to give a few recent
examples). The generic model developed and presented
here is based upon these prior works. This model
describes the following subsystems of EM:
(i) glycolysis (reduced to three reactions: vHK, vPFK
and vPK, see electronic supplementary material),
(ii) accumulation of fructose-2,6-bisphosphate (F26P),
an activator of glycolysis, through the vPFK2
reaction,
(iii) OP (vOP), regulated by the availability of PYR,
ADP and by the energetic state,
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(iv) PCr buffering with the creatine kinase (vCK)
reaction,
(v) lactate consumption or excretion through the
LAC dehydrogenase enzyme (vLDH) and external
supply of LAC (vLAC), and
(vi) adenylate kinase equilibrium (vADK), which
converts ADP to AMP þ ATP depending on
the energetic state of the system.
Parameters and steady-state concentrations used in
the generic model were taken (where available) from
the literature (see electronic supplementary material
for further details). Otherwise, curve-fitting techniques
were used to identify unknown parameters using the
procedures discussed in Cloutier et al. (in press) and
Schmidt & Jirstrand (2006). Mathematical representations for specific EM and EM regulation
mechanisms were also taken from the previously cited
modelling literature and used in the generic model.
The model was also validated against existing experimental and theoretical insights (see electronic
supplementary material), and is therefore believed to
produce a realistic behaviour for the generic dynamics
of EM.
Although the EM pathways are conserved between
organisms, the reactions may occur on significantly
different time scales. As an example, the metabolic
rate (i.e. moles of GLC consumed per unit of time)
will be 10 times lower in resting muscle cells than in
resting neurons. Accordingly, the metabolic rates of
various cells and bacteria will also be very different,
to the point that it would be impractical to directly
compare simulation results that are separated by several
orders of magnitude in time. Thus, in keeping with the
generic nature of the model and analysis, the time scale
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Control systems structures of EM
M. Cloutier and P. Wellstead
Table 1. Regulation mechanisms considered for EM. References: A, B: Lambeth & Kushmerick (2002) and references therein;
C: Heldt et al. (1972); D: Voet & Voet (2005) and references therein; E: Okar et al. (2001); F: Cloutier et al. (2009); Aubert &
Costalat (2005) and references therein.
regulation
observed in . . .
equation—reactions
A
vPFK regulation by ATP–AMP
concentrations
glycolytic organisms
vPFK is multiplied by
4
1 þ nATP ðATP=KI;ATP Þ 4
1 þ AMP=Ka;AMP
1 þ ðATP=KI;ATP Þ
1 þ nAMP ðAMP=Ka;AMP Þ
B
vPK regulation by ATP
glycolytic organisms
vPK is multiplied by
1 þ nATP ðATP=KI;ATP Þ 4
1 þ ðATP=KI;ATP Þ
C
vOP regulation by ADP/ATP
eukaryotes
vOP is multiplied by
1
1 þ nOP ðATP=ADPÞ
D PCr buffering
eukaryotes/animal
tissues
E
eukaryotes
vCK
PCr
ADP
F26P accumulation
[...]
Cr
ATP
F6P
+ v
PFK2
AMP
F
LAC shuttling
brain
vPFK
+
[...]
F26P
stimulation
neuronal EM
response
ATP
astrocytic LAC
production
used in this study will be arbitrary. However, this does
not affect the observations about the qualitative
description of each regulation mechanism. Moreover,
the time scale can be readily scaled up or down to represent a specific system, without materially affecting the
shape of the simulation results (i.e. reaction rates
expressed in s21 could be changed to min21 or h21).
One note of caution, however, is that care must be
taken to ensure that the model correctly reproduces
the specific properties of the system that is to be studied
and to note that reactions might not be scaled with
exactly the same factor. Despite this cautionary
remark on relative reaction rates, and as will be seen
in §3, the proposed control mechanisms seem to be
sufficiently robust to apply to a wide range of
conditions.
2.2. Regulation mechanisms
Table 1 presents the various regulation schemes we will
consider in this study. Note that for the EM implementations discussed in §3, the regulation mechanisms will
be added in a cumulative fashion from one scheme to
the next. Thus, the simulation of scheme E will also
include all the regulation mechanisms presented in
schemes A–D. On the other hand, schemes A–C
do not consider the action of PCr or LAC (which
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appear in schemes D and F, respectively). The only
exception will be case F, which will be shown to be a
very specific case of EM. This stage-wise addition of
regulation mechanisms allows the performance for various energy regulation schemes observed in living cells
to be built up from the most simple and common mechanisms of enzyme regulation (found in basically all
organisms) to the complex consideration of LAC
shuttling in the cerebral tissue (case F).
2.3. Dynamics of energy regulation
The model will be used to qualitatively analyse the
response of EM after a step change in energy demand.
Other studies could be performed, but this approach
of changing energy demand emulates many physiological systems. For example, muscle contraction or neuron
firing induces a sharp increase in ATP consumption.
The analysis of step changes in demand is also in keeping with the control systems theme of this paper.
Specifically, a standard way of assessing the performance
of a closed-loop control system is via its response to a
step change in demand (Seborg et al. 1989, ch. 3).
The step response gives both (i) a measure of the control system’s steady-state ability to meet changes in
demands and (ii) (via the nature of the transient
response) an indication of the stability properties of
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Control systems structures of EM
M. Cloutier and P. Wellstead
655
ATP (% of base value)
A
105
D
105
100
100
95
95
90
90
energy demand
(% of base value)
B
E
200
105
150
100
100
100
95
95
90
90
50
0
25
50
time
75
100
C
105
F
105
105
100
100
95
95
90
0
25
50
75
100 0
time
25
50
75
90
100
Figure 2. Performance of energy control—letters A –F refer to control schemes in table 1.
the feedback methods used (Seborg et al. 1989, ch. 11).
(We add, however, the caveat that these remarks refer
to the locally linearized responses of the system—we
defer more complex consideration of the nonlinear
aspects of EM response to a further work.)
Both steady-state and transient performance are
important, but in the dynamic environment of living
systems, the transient behaviour has special implications. In particular, the cell cycle implies continuous
variations in energy demand over time (i.e. the energy
demand is higher when the cells have to produce more
protein and cell components). In the same spirit,
many cellular functions, such as immunological
response, imply changes in energy demand over time
(Buttgereit et al. 2000). For these reasons, we analyse
the performance of EM regulation through its capacity
to cope with various changes in energy demand, and
relate this to the engineering control systems design
procedure of assessing controller performance using a
step response and/or disturbance rejection criterion.
3. SIMULATIONS RESULTS AND ANALYSIS
3.1. Comparing the different schemes for
ATP regulation
Figure 2 presents simulations results for the six different
regulation schemes (table 1) for a change in demand for
energy. Two possible changes in energy demand are
J. R. Soc. Interface (2010)
considered: (i) a step change of þ50 per cent in vATPase,
and (ii) a more gradual increase in the vATPase demand.
As can be seen from figure 2, the different control
schemes lead to qualitatively different behaviours. In
case A, the regulation is limited to vPFK and a sharp
increase in energy demand induces oscillations (grey
lines). Even if the transition is smooth (dark lines in
figure 2), the system tends to develop an oscillatory
behaviour. In control systems theoretic terms, and
when considered on a local linear basis, the multiplication of vPFK by the right-hand-column equation in
table 1 creates a negative feedback closed-loop system
on vPFK. Because of the large number of reactions in
the loop, and the associated lags and time delays, the
closed loop is unstable and an oscillatory response
ensues. If the closed-loop system were to be linear,
then the amplitude of theses oscillations would grow
exponentially without limit. However, the nonlinear
nature of the reactions and feedback ensure that the
oscillatory response grows into an oscillation with
bounded amplitude (technically termed a ‘limit cycle’;
Atherton 1982). This oscillatory response is an asymptotic property of the feedback loop and is independent
of the speed with which the change in energy demand
is made. This is illustrated in figure 2 by comparing
the responses to a rapid (step) change in demand with
the response to a slow change in demand. The step
demand causes the limit cycle to appear immediately,
whereas for a slow change in demand the amplitudes
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Control systems structures of EM
M. Cloutier and P. Wellstead
of the limit cycle oscillations grow more slowly over
time, but will eventually develop into stable sustained
oscillations of the same magnitude and frequency as
in the step-change case (simulations not shown).
The addition of regulation mechanisms in cases B – D
adds two further negative feedback loops within the
main (outer) loop provided by case A. These additional
loops can dampen or completely remove the oscillations
seen in case A. Notice that in cases A – D, although
the transient response to step demands improves
from case to case, the steady-state level of available
ATP decreases after the increase in vATPase demand.
This lack of adaptation is a known property of certain
control system structures where it is referred to as
‘steady-state error’ or ‘offset’ (Seborg et al. 1989,
ch. 8). It is thus relevant to observe that cases E and F
(in addition to good transient responses) also
exhibit perfect adaptation of the ATP level even in
the presence of a significant change in energy demand
(i.e. homeostasis). This observation tells us that
mechanisms E and F must add an additional control
mechanism to EM.
As mentioned previously, these observations of oscillation or perfect adaptation have all been reported
experimentally (Chance et al. 1964; Hochachka &
McClelland 1997), but previously they have not been systematically associated with the control system structures
within the EM. Our modelling framework addresses this
issue, and is able to clearly separate the contribution of
each EM component to the observed systems behaviour.
In the following sections, each reaction scheme will
be studied in more detail and with reference to the
corresponding control components.
3.2. Putting the ‘control’ back in metabolic
control
The cases presented in figure 2 will be analysed in
§§3.3 –3.7 from a control theoretic view. First, cases
A – C show regulation of reaction rates by the energetic
state. As will be shown, this is equivalent to proportional feedback control (i.e. reaction rates
producing energy are proportional to the offset in
ATP). In case D, the addition of the dynamic contribution of a buffer for the energy balance shows
(figure 2 ), under specific conditions, similarities found
when derivative control is added to proportional feedback. A further level of sophistication comes with the
accumulation of an activator of glycolysis, such as
F26P (case E). This provides additional control in the
form of an integral action, as the accumulation occurs
when the energy level is perturbed. Finally, a very
specific case is observed in figure 2 for case F. In this
situation, the increase in energy demand is forwarded
to a second system, which responds by producing
energy substrate (LAC) for the first system. As will be
shown, this is identical to feed-forward control.
3.3. Proportional and cascade control through
enzyme regulation
Figure 3 shows a breakdown of cases A – C, where different schemes for enzyme regulation were considered.
J. R. Soc. Interface (2010)
First, all the regulation terms considered are proportional to ATP (or ATP/ADP and AMP, other
indicators of energy level). As these reactions ultimately
lead to energy production, a striking analogy with proportional control can be drawn. Figure 3 reveals the
negative feedback structures in EM in a diagrammatic
form. In particular, the figure shows how the information about the state of the system (ATP, ADP) is
sent back to modulate the various reactions of EM.
As can be seen from the regulation expressions, the reaction rates are proportional to the energetic state,
although here the relations are nonlinear. An important
observation here, though, is that all the nonlinear
expressions work in the same ‘direction’. When
the energetic state is reduced (ATP#, ATP/ADP#),
reaction rates are increased in order to produce
more energy (e.g. proportional negative feedback
action).
In case A (blue lines), only the first reaction (vPFK) is
controlled. A delay is thus observed before the activation of the last step (blue line for vOP). This is
perfectly normal, as the system has to accumulate
metabolites (GAP, PYR) in order to activate the subsequent steps. As a result, an oscillatory behaviour is
induced which is typical of proportional feedback control with significant transport delay (Marshall 1979).
The reduction and eventual removal of the oscillations moving from case A to C is analogous to the
control systems technique of cascade control (Seborg
et al. 1989, ch. 18). Cascade control is used in corresponding chemical process control systems, in which
there are a number of stages (and corresponding
delays) between the point at which demands are made
and where the control signal is applied. Cascade control
consists of introducing additional inner negative feedback loops to intermediate stages, just as occurs in
cases B and C. The concept of simultaneous activation
of reactions is very well known in biological systems. We
see here that it applies to EM with a striking similarity
to cascade control and it potentially provides a new way
to assess how oscillations might develop (or be
surpressed) in EM.
A thorough analysis of oscillatory behaviour in glycolysis (without OP) can be found in Termonia &
Ross (1981) and Heinrich & Schuster (1996) or, more
recently, Chandra et al. (2009). In these studies, the
oscillatory behaviour is usually explained through a
mix of autocatalysis (the pathway uses energy to produce more energy) and feedback (high-energy state
inhibits the pathway). In this study, we do account
for the autocatalytic nature of the pathway: the first
two steps (vHK, vPFK) consume energy and the last
two (vPK, vOP) produce more energy. However, the
high ‘gain’ in energy production by vOP completely
changes the nature of the system, when compared
with glycolysis alone. In processing one molecule of
GLC, glycolysis uses 2 ATP in the first steps to produce
4 ATP in the end, while OP will yield 30 molecules of
ATP per molecule of GLC (i.e. the gain is 15 times
higher). This higher gain in the last step (vOP) exacerbates the importance of the delay and thus hints at
an alternative explanation—proportional control with
delay—for the appearance of oscillations in EM.
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reaction rates
(% of base value)
Control systems structures of EM
200
200
200
150
150
150
100
50
0
10
20
vPK
10
20
50
30
0
vOP
10
20
30
ADP
GLC
vPFK
GAP
vPK
PYR
200
energy demand
150
vOP
100
ATP
50
0
105
1+nATP·(ATP/Ki)
1+(ATP/Ki)
4
1+nATP·(ATP/Ki)
1+(ATP/Ki)
657
100
100
vPFK
50
30
0
M. Cloutier and P. Wellstead
4
1
1+nOP·(ATP/ADP)
10
20
time
30
ATP
100
95
90
0
10
20
time
30
Figure 3. Regulation of EM reactions with cascade structure. States and reactions are per cent of steady-state values.
Blue lines, A; green lines, B; red lines, C.
However, an important observation here is that glycolytic oscillations are usually not observed in aerobic
conditions in yeasts (a eukaryote). As reviewed in
Richard (2003), anaerobic conditions must usually be
imposed to induce glycolytic oscillations. Interestingly,
our framework is completely coherent with this observation. If we model a system with proper mitochondrial
regulation, as in case C (representative of aerobic conditions in eukaryotes), we do not observe oscillations.
In this situation, the reactions of OP are enclosed in
the mitochondria, whose overall activity is known to be
regulated by the ATP/ADP ratio (Heldt et al. 1972).
In our control framework, proportional feedback is introduced around the OP process, thus the delay before ATP
production is reduced and oscillations are completely
removed (red lines in figure 3). Simulations (not presented here) also show that this ‘eukaryotic’ aerobic
system (case C) is extremely robust under a wide range
of stimulations, as it can sustain 5- to 10-fold increases
in energy demand without entering unstable regimes.
As our modelling and control framework can give
some physiological insights on the development or
removal of oscillations in EM, it could be used to complement previous studies on this problem. The possible
advantage of the oscillatory behaviour in glycolysis
was mentioned by Selkov (1980) and Richter & Ross
(1981). A study by Chandra et al. (2009) also mentions
that glycolysis might operate near oscillatory regimes
for performance and trade-off reasons. However, oscillations are not necessary or even detrimental to
certain tissues, such as the heart (O’Rourke et al.
1994), and this shows that mechanisms that remove
or create oscillations also have an important role.
Finally, a steady-state offset in ATP is observed
in cases A – C. This is also observed in proportional
J. R. Soc. Interface (2010)
control and cannot be removed by cascade action.
Such controllers will always have some error between
the demanded output value and the actual value. Interestingly, as a change in metabolites (here ATP, ADP) is
required to have a change in flux, the proportional control analogy is coherent with the metabolic control
analysis (MCA) framework (Fell 1997; Kacser &
Burns 1973), where elasticity coefficients are a measure
of the change in flux after a change in metabolite. However, MCA was initially developed as a steady-state
analysis and thus it would not necessarily differentiate
between the three cases presented in A – C, as the different systems could reach the same steady state, but
through different time courses. Nevertheless, the
analogy with MCA is insightful, as the elasticity coefficients in the MCA framework would give a local
linearization of the proportional gains in our framework. Extensions to the MCA framework that account
for transient states (Heinrich & Reder 1991) could
also be considered for comparison with the control
framework presented here.
3.4. Derivative action through temporary
energy buffers
Derivative control action is used in control theory to
help reduce oscillatory behaviour (Seborg et al. 1989,
ch. 8). By including a control action that is proportional
to the derivative of the output, the control action has
some predictive compensation for rapid changes in the
output variable. In the case of EM, the addition of the
PCr buffer (case D) smooths the transition after a
sharp increase in energy demand (when compared with
case C). As can be seen for case D in figure 2 (grey
lines), this action removes undershoot in ATP (observed
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105
ATP
100
energy demand
95
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0.5
vCK versus dATP/dt
0.4
0
50
0
10
20
30
10
20
30
0.2
vCK
200
0
vCK
–0.2
–0.04
0.3
time
–0.02
0
dATP/dt
0.02
0.1
–0.1
0
10
20
30
time
Figure 4. Derivative action from the PCr buffer. States and reactions are per cent of steady-state values, except for vCK
(mM unit time). Black lines, case C; grey lines, case D.
in case C). This effect, which is the desired result of
derivative action, is discussed in more detail here.
First, we consider the stoichiometry of the conversion
between PCr and Cr:
PCr þ ADP $ Cr þ ATP:
ð3:2Þ
The reaction is reversible and reaches equilibrium
when the ATP/ADP ratio is constant. As PCr and Cr
are not involved in other reactions, they only respond
to changes in ATP/ADP. When the ATP/ADP ratio
is perturbed by a sudden increase in energy demand,
the PCr/Cr equilibrium will be changed. Specifically,
it is possible to derive the action from vCK when ATP
changes over time. If we assume that a certain
amount, ‘d ’ of ATP is consumed within a time interval
(Dt), we can re-write the expression for vCK accordingly
vCKjt¼Dt ¼ kf;CK PCr ðADP þ dÞ kr;CK Cr
ðATP dÞ:
ð3:3Þ
Here we neglect the fact that a very small amount of
AMP will also be formed, as it is 100 times lower than
the amount of ATP. It is thus possible to separate the
vCK reaction in two terms. The first depends on the
states and reaction constants (PCr, ATP, kf,CK . . .)
and the second depends also on the derivative (d )
J. R. Soc. Interface (2010)
vCKjt¼Dt ¼ ½kf;CK PCr ADP kr;CK Cr ATP
þ ½kf;CK PCr þ kr;CK Cr d:
ð3:4Þ
ð3:1Þ
The conversion of one molecule of PCr to Cr allows
the regeneration of one molecule of ATP from ADP
without contribution from other cellular subsystems.
Obviously, this action is limited by the amount of available PCr, hence its role is to act as a temporary energy
buffer. Conversely, the PCr can be replenished if the
energetic state (ATP/ADP) is favourable. The conversion of PCr to Cr is catalysed by the enzyme creatine
kinase (vCK). The regulation of this enzyme’s activity
can be described by mass action kinetics
vCK ¼ kf;CK PCr ADP kr;CK Cr ATP:
that was imposed
As can be seen from equation (3.3), the term with the
derivative will be positive, that is, a decrease in ATP
will inevitably lead to ATP regeneration by the vCK
reaction. This action is potentially nonlinear, as the
dynamics of the first term on the right-hand side is
not necessarily negligible. Simulation results for this
system, with comparison to a system without PCr, are
presented in figure 4.
As can be seen, the effect of vCK leads to energy
production at a rate proportional to the derivative of
ATP (grey lines in figure 4). The effect is almost
linear over a change of 100 per cent in ATP demand.
However, it must be mentioned here that this linearity
is highly dependent on the parameters of the creatine
kinase system (simulation results not shown). However,
under reasonable hypotheses, we see that the effect can
show similarities with derivative control.
Interestingly, phosphagens for transient energy
storage are found mostly in cells and tissue that have
the ability to produce mechanical work, that is, cells
that use flagella for propulsion, animal muscle tissues,
insect tissues etc. This observation obviously hints at
a possible action for phosphagens in the transition
from low- to high-energy demand. Here we show that
this effect could be, under certain conditions, similar
to a derivative control component. As mentioned earlier, this is not incompatible with the role of PCr as a
spatial buffer, when PCr and Cr can be spatially organized or compartmentalized to shuttle energy from
the mitochondria to the site of energy consumption
(Bessman & Geiger 1981). In the simulation of
figure 4, where a 100 per cent change in ATP demand
was imposed, the PCr concentration was reduced by
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Control systems structures of EM
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M. Cloutier and P. Wellstead
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30
60
250
90
ADP
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+
F6P
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150
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(% of base value)
100
500
150
0
250
0
30
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60
90
30
–100
0
30
time
60
105
750
vPFK2
250
50
0
60
90
60
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200
F26P
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ATP
ADK
eq.
vPFK2
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200
vOP
...
F26P
350
659
90
0
AMP
ATP
100
95
30
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60
90
90
0
30
time
Figure 5. Integral action in EM. States and reaction rates are per cent of steady-state values. Black lines, vPFK ¼ 0; grey lines,
vPFK / AMP.
only 3 mM (simulation not shown), whereas muscle
tissues often contain PCr at concentrations up to
35– 40 mM. Thus, there is potentially enough PCr in
muscle to perform both actions (spatial and temporal
buffer).
Finally, in some tissues, the change in energy
demand can be up to 50-fold (Liguzinski & Korzniewski
2006), and this would require a high buffering capacity
from PCr. However, cells and tissues have other
mechanisms to reduce the changes in ATP.
3.5. Integral feedback in energy metabolism: the
forgotten side reaction
Glycolysis is usually presented as a linear pathway with
nine reactions. However, this representation neglects a
very important side reaction, the PFK2 (vPFK2 in
figures 1 and 5). This reaction allows the accumulation
of F26P, one of the strongest activators of glycolysis
(it activates vPFK). F26P and its accumulation
through the PFK2 reaction is an important regulation
mechanism for the glycolytic flux (Pilkis 1990). The
enzyme catalysing the vPFK2 reaction is bi-functional
and can catalyse the reverse reaction (F26P to F6P;
figure 1). As we will show here, this side reaction
with the accumulation of an activator produces,
under certain conditions, an integral feedback action
on EM. And it is the integral action that can
remove steady-state offset and provide perfect
adaptation.
First, during an energy stress, the AK equilibrium
(figure 1 and electronic supplementary material)
produces AMP proportionally to the reduction in
ATP concentration from the steady state (i.e. a few
J. R. Soc. Interface (2010)
per cent drop in ATP can lead to a several fold increase
in AMP). Thus, AMP is proportional to the ‘error’ in
ATP (i.e. the difference from the steady state)
½AMP / error:
ð3:5Þ
AMP then activates the AMP-dependent protein kinase
(AMPK), which in turn activates vPFK2 (Hardie 2004).
In this study, we assume the dynamics of AMPK can be
neglected, and so we draw a direct relationship between
AMP and vPFK2. This simply assumes a fast equilibrium
between AMPK phosphorylation state and AMP concentration. Thus, vPFK2 is proportional to AMP, and
the accumulation of F26P will be described as follows:
dF26P
¼ vPFK2 / AMP:
dt
ð3:6Þ
Thus, the amount of F26P present in the system will be
proportional to the integral of AMP, that is, the integral of the error on ATP. As F26P activates a key
step of glycolysis (vPFK), we can draw the following
relationship:
ð
ð3:7Þ
vPFK / F26P / error:
t
And the integral feedback action of F26P thus becomes
obvious. A similar identification of integral control
(with its concomitant ability for perfect adaptation)
has been made previously for other biological systems,
with some examples found in Csete & Doyle (2002) or
El-Samad et al. (2002).
Figure 5 shows simulation results in EM with and
without the integral control mechanism (expanded
from cases E and D in figure 2). As can be seen in
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M. Cloutier and P. Wellstead
figure 5, F26P can indeed be accumulated during
energetic stress to activate glycolysis and produce
homeostatic behaviour (see grey line for ATP in
figure 5). The accumulation of F26P allows EM to be
maintained in an active state, even though the ATP
level is back to its original state, something that is technically impossible with enzyme regulation alone (see
cases A – D). As was mentioned in a review by Okar
et al. (2001), the PFK2 enzyme and action of F26P
are found in all eukaryotes (but not in prokaryotes)
and their roles in adaptation and homeostasis under
changing external conditions are acknowledged. Interestingly, this mechanism would provide eukaryotes
with a definite advantage over prokaryotes for the
precise regulation of the glycolytic flux under varying
energy loads.
Integral action is essential when perfect adaptation
(e.g. zero steady-state error) is required. However, as
can be seen in figure 5, the effect of integral action on
vPFK can be destabilizing, as a small, damped oscillation
is easily induced. Depending on the specific parameters
of the system, the integral effect can also lead to
unstable behaviour (simulations not shown). This is primarily because integral action depends upon the sum of
past information, thus introducing a destabilizing
effect. In addition, and as was the case for proportional
control, the integral action is performed on the
upstream part of glycolysis, and because delays are
unavoidable before the activation of vOP and energy
production, there is a further tendency for it to
destabilize the loop.
In addition to giving perfect adaptation, there is
another reason why it is advantageous to have the integral action on PFK. The ‘pace-making’ role of PFK in
glycolysis is well known (Westerhoff 1995) and observations within the MCA framework also show that
the enzyme PFK is highly coupled to energy demand
(Thomas & Fell 1998). Thus, under various conditions,
PFK controls the amount of substrate that enters the
glycolytic system. Having integral action on this
enzyme provides again an undisputable advantage, as
it is the enzyme that has to be maintained in its
active state during high demand periods.
Finally, it must be noted that many cellular stress
signals affect AMPK and thus potentially affect vPFK2
and F26P dynamics. As regards energy stresses, it is
known that F26P accumulation can occur because of
hormonal regulation during exercise (Winder & Duan
1992). As mentioned by Holloszy (2008), AMPK is
involved in short-term, biochemical adaptation (such
as modelled here), but it is also involved in long-term
adaptation to energy stresses (i.e. increased GLC transport capacity and increased mitochondrial biogenesis).
Thus, although we concentrate our study here on only
one possible role for F26P in biochemical regulation,
with only one activation mechanism (i.e. AMPK signalling because of high AMP), it must be kept in mind that
the integration of energetic stresses is much more broad
and not yet fully understood (Hardie 2007). However,
this complexity in cellular signals is also an argument
for a rigorous, dynamic systems analysis, and the
control framework presented here could be used to
that end.
J. R. Soc. Interface (2010)
3.6. Feed-forward control: astrocyte– neuron
coordination
The brain is probably the most complex tissue in the
animal kingdom, and precise functioning and regulation
of EM in the cerebral tissue is an area of research where
many questions are yet to be answered. One of these
questions regards the possible release of LAC by
astrocytes to fuel EM in neurons. Since the astrocyte –
neuron lactate shuttle (ANLS) hypothesis was put
forward 15 years ago (Pellerin & Magistretti 1994),
many studies have been performed, but with mixed
results. For example, it was shown that cultured neurons can use LAC to fuel OP (Dringen et al. 1993)
and, as mentioned earlier, it is also known that LAC
can regulate ATP concentration in neurons (Ainscow
et al. 2002). However, other studies and reviews raise
doubt about the exact role of LAC in brain EM (Chih
et al. 2001; Fillenz 2005), as neurons also use GLC to
fuel their energy needs.
An important factor to consider in the role of LAC is
the presence of glial cells (astrocytes) in the cerebral
tissue. As part of the ANLS hypothesis, astrocytes are
activated in coordination with neurons as excess neurotransmitters released by neurons during high-activity
periods are taken up by glial cells. In order to perform
this task, astrocytes need energy and they consequently
undergo an increase in glycolytic rate and produce
LAC. Measurements in the cerebral interstitial fluid
during high-activity periods show a small initial dip
(5%) in LAC concentration, followed by a significant
increase (greater than 50%; Hu & Wilson 1997;
Lowry & O’Neill 2006). This implies a possible mechanism where LAC is being consumed by neurons and
produced by astrocytes at a higher rate but after an
initial delay.
The presence of a secondary cell system (astrocytes)
that produces energy substrate for the main functional
system (neurons) is analogous to the feed-forward control action used in engineering control systems
(Seborg et al. 1989, ch. 17). The aim of feed-forward
control is to give an early and rapid response to a fast
change in demand in a way that cannot be met by integral feedback action that must pass through a number
of intermediate states. Thus, in the brain, EM information on the ‘disturbance’ in energy demand (i.e.
increase in neurotransmitter circulation) is forwarded
to astrocytes, which respond by producing LAC. The
higher LAC concentration then favours uptake and oxidation in neurons and improves ATP regulation. This
mechanism does not require any feedback action
within neuronal metabolism (i.e. LAC will increase
regardless of ATP levels in neurons), hence the feedforward analogy. In classical control, this type of
controller can be finely tuned to reject disturbances
on the system. Again, analogies are also found in the
systems biology literature, as this mechanism can be
compared with the perfect adaptation system presented
in Tyson et al. (2003).
Simulation results with this feed-forward LAC
mechanism and a comparison to integral feedback are
presented in figure 6. Two successive and short pulses
in energy demand were implemented. The change in
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M. Cloutier and P. Wellstead
661
140
LAC
astrocytic
metabolism
LAC production
125
feed-forward scheme
energy demand signal is
carried to an external
system (astrocytes)
110
95
glycolysis
150
60
vLDH
vPFK
125
PYR
time
180
120
120
oxidative phosphorylation
150
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ADP
90
125
100
100
75
135 energy demand
0
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60
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120 180
0
60
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103
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180
ATP
AMP
125
ATP
100
97
100
75
60
pulses in energy demand
(neuronal activity)
+
150
0
0
60 120
time
180
integral feedback scheme
energy stress activates glycolysis
94
0
60
120
time
180
Figure 6. Comparison of integral feedback (blue lines) and feed-forward (green lines) for rapid transitions in energy demand.
States and reaction rates are per cent of steady-state values.
energy demand here is relatively small (25%) as it is
reported that the basal metabolic rate in neurons is relatively important (i.e. as much as 90% of the active
rate). In a previous study (Cloutier et al. in press), we
observed that, during physiological stimulations, a
maximum of 20– 25% increase in energy demand
could be observed. The input used in figure 6 thus
allows an assessment of the capacity of the two control
mechanisms (integral: blue lines; feed-forward: green
lines) to adapt to repetitive bursts in activity, a control
problem faced by neurons that must rapidly switch
between the resting state and stimulation. Also note
that our simulation qualitatively reproduces the LAC
profiles mentioned previously, with a small initial dip
followed by an overshoot.
As seen from the right-hand column of figure 6, the
feed-forward mechanism with LAC production by an
external system significantly improves the energy regulation. The changes in ATP are not large (less than 5%)
in both the integral (blue lines) and feed-forward cases
(green lines). However, the feed-forward scheme still
reduces the ATP variations threefold (as compared
with integral action) and produces a more stable behaviour. From a control theoretic viewpoint, this is to be
expected, because the major problem of integral control
is that it is slow to respond to changes in demand on the
system, as the whole glycolytic pathway must be activated during stimulation and de-activated after the
stimulus is removed. In control terminology, the integrators ‘wind up’ to compensate for a demand and
then take time to ‘un-wind’ after the event (Seborg
et al. 1989, ch. 8). Even though we have used a simplified, shorter pathway to describe glycolysis, it is always
slower than the direct feeding of vOP with a sharp
increase in LAC concentration (green lines for vOP
and LAC in figure 6).
J. R. Soc. Interface (2010)
The superior performance of a feed-forward strategy
occurs because it has no memory of the past and
therefore responds instantaneously to demands—a
very important feature in neuronal energy regulation.
This point raises questions about the relevance of integral feedback in neuronal EM. As can be seen, the use of
integral feedback would potentially be destabilizing
for neurons submitted to short bursts in energy
demand—the slow recovery of integral action would
also introduce the potential of energy stress within the
brain EM. It is therefore interesting to note that neuronal cells are the only eukaryotic cells reported to
have no PFK2 activity (Bolanos et al. 2004; Rider
et al. 2004). Thus, they would be the only cells in
animal tissues not to rely on the integral action of
F26P in their EM structure. As we can see from the control perspective, a neuronal EM structure where the
integral action is replaced by a well-tuned feed-forward
pathway would indeed improve ATP regulation.
Finally, the performance of this feed-forward system
also depends on the specific parameters used in the
modelling. Here, the modelling is a simplified version
of a model for the brain EM, and it reproduces LAC
profiles measured during sensory stimulations (see
Cloutier et al. in press for further details). From this
increase in LAC shuttling, the OP regulation can produce a fast and practically perfect adaptation of ATP
levels in neurons. Thus, it seems that the reported
LAC kinetics in the cerebral tissue do hint at a
well-adapted feed-forward mechanism.
4. CONCLUSION
4.1. Control structure of energy metabolism
The main result of this study is a delineation of the
metabolic regulation of EM in terms of basic control
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Control systems structures of EM
M. Cloutier and P. Wellstead
control
mechanisms
eATP
enzyme regulation
e
t
u
eATP
nPFK
nOP
nCK
metabolite
accumulation
d(eATP)
dt
PCr buffering
AMP
ATP
ADP
(a)
energy
load
nATPases
+
yss
u
e
y
[ATP]
C
S
regulation
mechanisms
glycolysis
OP
–
[ATP]ss
(b)
energy
load
control structures
nATPases
yss
[ATP]ss
+
–
glycolytic
regulation
+
S1
C
S2
–
OP
glycolysis
y
[ATP]
Cc
OP
regulation
(c)
lactate
shuttle
yss
[ATP]ss
+
–
C
S1
glycolytic
regulation
glycolysis
CFF
energy
load
nATPases
+
+
S2
–
y
[ATP]
OP
Cc
OP
regulation
Figure 7. Mechanisms and control structures for EM. (a) Case ‘A’ is a general structure for a system (S), with input (u) managed
by a controller (C). (b) Case ‘B’ shows cascade control (CC) to speed up the response. (c) Case ‘C’ shows feed-forward control
(CFF) where a change in the load is used directly for fast regulation. Biological mechanisms are shown in grey.
systems concepts. This was achieved through a dynamic
analysis of energy regulation using a generic model for
EM. As the model was built upon reported mechanisms
in EM, our analysis allows a systems understanding of
the problem of energy regulation in biological systems.
Interestingly, strong evidence was found for the use
within EM of the most widely used forms of control
components ( proportional, derivative, integral) and
control structures (feedback, feed-forward and cascade
J. R. Soc. Interface (2010)
control). These mechanisms and structures are
summarized in figure 7, where the correspondence
between control mechanisms and structures and their
biophysical equivalents (text in grey) is shown.
Similar analogies between biological regulation and
classical control have been reported in other studies,
but, to the best of our knowledge, this is the first extensive study of such basic mechanisms and structures in
EM. Within this framework, it was observed that cells
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Control systems structures of EM
with specific requirements for EM regulation will rely on
the appropriate control structure to reduce ATP variations. This observation even holds for the regulation
of energy in the brain, where two cell types (neurons
and astrocytes) will cooperate through a tuned feedforward mechanism in order to reduce ATP variations
in the cells that have to perform a critical function.
Comparisons with experimental works and theoretical
frameworks (such as MCA) also showed that a dynamic
control analysis is totally coherent with the current
knowledge on biological regulation. In this regard, our
framework was able to reproduce the most common systems properties observed in EM, such as oscillations
and homeostasis. Our approach thus offers added
insight into, and a reappraisal of, known mechanisms
for energy regulation.
4.2. Final comments
This study was focused on EM as a means of drawing
analogies between biological mechanisms and control.
In this spirit, the results obtained here cover the most
common control components (PID) and structures
(feedback, feed-forward and cascade control). These
components and structures are fundamental to all
dynamical control, such that the control framework presented here could eventually be extrapolated to other
biological reaction systems. In this spirit, the proper
classification of biological mechanisms in terms of corresponding control components and structures should
help increase the appreciation of the role of the control
theory in biological systems. For example, the basic
controller components—proportional, derivative and
integral—have quite specific control functions and corresponding advantages and limitations in their use. The
ability to identify biological objects that perform
these functions can therefore help us understand the
capabilities and constraints of a biological system—
just as it does in the analysis of physical systems.
We also re-emphasize that our description has been
specifically aimed at describing the core control mechanisms and structures in EM. The description given
here can be built upon to add additional substrates,
reactions and more complex interactions. Such
additions will modify the performance of the resulting
control system, but the function of the core PID mechanisms and feedback/feed-forward structures will not
be altered. We see therefore the control blocks and
structures described here as offering the core framework
for understanding the dynamic regulation of metabolism in general terms; a framework to which further
additional control functions can be added.
The idea of comparing the dynamic regulation
requirements of a biological system with potential
control components could also be used to classify and
analyse regulation mechanisms within a more rigorous
framework. For example, cascade control is an
expression of the more general control concept of state
feedback. We therefore anticipate that within a control
theoretic framework it will be possible to produce a systematic analysis using tools from control engineering,
such as stability analysis, robustness etc., including a
full consideration of the nonlinear nature of metabolic
J. R. Soc. Interface (2010)
M. Cloutier and P. Wellstead
663
feedback that we (for simplicity) have avoided in this
study. For a wider discussion of these issues and the
potential role of control and feedback in biology, see
Wellstead et al. (2008).
The authors wish to thank the anonymous reviewers for their
insightful suggestions and references, and Rick Middleton for
his additional insights on the control mechanisms described in
this work. Financial support from the Science Foundation of
Ireland (03/RP1/I382) is gratefully acknowledged.
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